x PREFACE

graduate student of today must, as soon as possible, get a thorough training
in this abstract and axiomatic way of thinking, if he is ever to understand what
is currently going on in mathematical research. This volume aims to help the
student to build up this " intuition of the abstract" which is so essential in the
mind of a modern mathematician.

It is clear that students must have a good working knowledge of classical
analysis before approaching this course. From the strictly logical point of
view, however, the exposition is not based on any previous knowledge, with
the exception of:

1. The first rules of mathematical logic, mathematical induction, and the
fundamental properties of (positive and negative) integers.

2. Elementary linear algebra (over a field) for which the reader may consult
Halmos [11], Jacobson [13], or Bourbaki [4]; these books, however, contain
much more material than we will actually need (for instance we shall not use
the theory of duality and the reader will know enough if he is familiar with the
notions of vector subspace, hyperplane, direct sum, linear mapping, linear
form, dimension, and codimension).

In the proof of each statement, we rely exclusively on the axioms and on
theorems already proved in the text, with the two exceptions just mentioned.
This rigorous sequence of logical steps is somewhat relaxed in the examples
and problems, where we will often apply definitions or results which have not
yet been (or ever will never be) proved in the text.

There is certainly room for a wide divergence of opinion as to what parts of
analysis a student should learn during his first graduate year. Since we wanted
to keep the contents of this book within the limits of what can materially be
taught during a single academic year, some topics had to be eliminated.
Certain topics were not included because they are too specialized, others
because they may require more mathematical maturity than can usually be ex-
pected of a first-year graduate student or because the material has undoubtedly
been covered in advanced calculus courses. If we were to propose a general
program of graduate study for mathematicians we would recommend that
every graduate student should be expected to be familiar with the contents of
this book, whatever his future field of specialization may be.

I would like to express my gratitude to the mathematicians who have
helped me in preparing these lectures, especially to H. Cartan and N. Bour-
baki, who allowed me access to unpublished lecture notes and manuscripts,
which greatly influenced the final form of this book. My best thanks also go
to my colleagues in the Mathematics Department of Northwestern University,
who made it possible for me to teach this course along the lines I had planned
and greatly encouraged me with their constructive criticism.

April, 1960 J. DIEUDONN&ntain definitions and results that have not up to that point appeared in